Motivated by the interest in the role of stiff polymers like actin in cellular processes and as components of biopolymer networks like the cytoskeleton, we present a statistical mechanical study of the twist elasticity of stiff polymers. We obtain simple, approximate analytical forms for the writhe distribution at zero applied force. We also derive simple analytical expressions for the torque-twist relation and discuss buckling of stiff polymers due to the applied torques. The theoretical predictions presented here can be tested against single-molecule experiments on neurofilaments and cytoskeletal filaments like actin and microtubules.